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# Polytope of Type {4,26}

Atlas Canonical Name : {4,26}*208
Also Known As : {4,26|2}. if this polytope has another name.
Group : SmallGroup(208,39)
Rank : 3
Schlafli Type : {4,26}
Number of vertices, edges, etc : 4, 52, 26
Order of s0s1s2 : 52
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,26,2} of size 416
{4,26,4} of size 832
{4,26,6} of size 1248
{4,26,8} of size 1664
Vertex Figure Of :
{2,4,26} of size 416
{4,4,26} of size 832
{6,4,26} of size 1248
{3,4,26} of size 1248
{8,4,26} of size 1664
{8,4,26} of size 1664
{4,4,26} of size 1664
{6,4,26} of size 1872
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,26}*104
4-fold quotients : {2,13}*52
13-fold quotients : {4,2}*16
26-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,52}*416, {8,26}*416
3-fold covers : {12,26}*624, {4,78}*624a
4-fold covers : {4,104}*832a, {4,52}*832, {4,104}*832b, {8,52}*832a, {8,52}*832b, {16,26}*832
5-fold covers : {20,26}*1040, {4,130}*1040
6-fold covers : {24,26}*1248, {12,52}*1248, {4,156}*1248a, {8,78}*1248
7-fold covers : {28,26}*1456, {4,182}*1456
8-fold covers : {8,52}*1664a, {4,104}*1664a, {8,104}*1664a, {8,104}*1664b, {8,104}*1664c, {8,104}*1664d, {16,52}*1664a, {4,208}*1664a, {16,52}*1664b, {4,208}*1664b, {4,52}*1664, {4,104}*1664b, {8,52}*1664b, {32,26}*1664
9-fold covers : {36,26}*1872, {4,234}*1872a, {12,78}*1872a, {12,78}*1872b, {12,78}*1872c, {4,78}*1872
Permutation Representation (GAP) :
```s0 := (27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)
(37,50)(38,51)(39,52);;
s1 := ( 1,27)( 2,39)( 3,38)( 4,37)( 5,36)( 6,35)( 7,34)( 8,33)( 9,32)(10,31)
(11,30)(12,29)(13,28)(14,40)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)
(22,45)(23,44)(24,43)(25,42)(26,41);;
s2 := ( 1, 2)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(14,15)(16,26)(17,25)(18,24)
(19,23)(20,22)(27,28)(29,39)(30,38)(31,37)(32,36)(33,35)(40,41)(42,52)(43,51)
(44,50)(45,49)(46,48);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(52)!(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)
(36,49)(37,50)(38,51)(39,52);
s1 := Sym(52)!( 1,27)( 2,39)( 3,38)( 4,37)( 5,36)( 6,35)( 7,34)( 8,33)( 9,32)
(10,31)(11,30)(12,29)(13,28)(14,40)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)
(21,46)(22,45)(23,44)(24,43)(25,42)(26,41);
s2 := Sym(52)!( 1, 2)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(14,15)(16,26)(17,25)
(18,24)(19,23)(20,22)(27,28)(29,39)(30,38)(31,37)(32,36)(33,35)(40,41)(42,52)
(43,51)(44,50)(45,49)(46,48);
poly := sub<Sym(52)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope